## Archival Version

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The incipient infinite cluster in high-dimensional percolation
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## Tarski's problem about the elementary theory of free
groups has a positive solution

### Olga Kharlampovich, Alexei Myasnikov

**Abstract.**
We prove that the elementary theories of all nonabelian free groups
coincide and that the elementary theory of a free group is decidable.
These results answer two old questions that were raised by A. Tarski
around 1945.

*Copyright 1998 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**04** (1998), pp. 101-108
- Publisher Identifier: S 1079-6762(98)00047-X
- 1991
*Mathematics Subject Classification*. Primary 20E05, 20F10
*Key words and phrases*.Free group, elementary theory, Tarski
- Received by the editors May 25, 1998
- Posted on December 14, 1998
- Communicated by Efim Zelmanov
- Comments (When Available)

**Olga Kharlampovich**

Department of Mathematics and Statistics,
McGill University,
805 Sherbrooke St. West,
Montreal, QC, Canada H3A 2K6

*E-mail address:* `olga@Math.McGill.CA`

**Alexei Myasnikov**

Department of Mathematics,
City College,
Convent Ave. \& 138th St.,
New York, NY 10031

*E-mail address:* `Alexei@rio.sci.ccny.cuny.edu`

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